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Heterogeneous FirmsNotes for Graduate Trade Course
J. Peter Neary
University of Oxford
January 30, 2013
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 1 / 29
Plan of Lectures
1 Empirical Background
2 Overview of the Melitz Model
3 Equilibrium in Autarky
4 Effects of Trade
5 Extensions
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 2 / 29
Empirical Background
Plan of Lectures
1 Empirical Background
2 Overview of the Melitz Model
3 Equilibrium in Autarky
4 Effects of Trade
5 Extensions
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 3 / 29
Empirical Background
Empirical Background
The Data Revolution: Micro-Data on Firms.
Evidence totally orthogonal to traditional theory:
Exporting firms are:
Rare!: Very few firms export;... and those that do sell most of their output domestically;
Larger;More productive;... ex ante (”selection into exporting”), not ex post (”learning by
exporting”) (Clerides et al. QJE 1995; Bernard-Jensen JIE 1999).OlderPay higher wages
Effects of trade liberalisation:
Forces least productive firms to exit (Bernard and Jensen, JIE 1999).Encourages market share reallocation towards more productive firms;... and so raises aggregate productivity (Pavcnik REStud 1999,
Bernard-Jensen-Schott JIE 2006).
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 4 / 29
Overview of the Melitz Model
Plan of Lectures
1 Empirical Background
2 Overview of the Melitz ModelDynamic Industry EquilibriumContinuum CES Preferences
3 Equilibrium in Autarky
4 Effects of Trade
5 Extensions
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 5 / 29
Overview of the Melitz Model Dynamic Industry Equilibrium
Dynamic Industry Equilibrium
Monopolistic competition with CES preferences; so what’s new?
Melitz (2003), Helpman-Melitz-Yeaple (2004) [HMY]
Population of ex ante identical firms.
Firms face two sources of uncertainty:1 Uncertain productivity ϕ / cost c ; quality another interpretation.
Drawn from a known distribution with pdf g(
1c
)with positive support
over (0, ∞) and associated cdf G(
1c
). [g
(1c
)= G ′
(1c
)]
To learn its c, a firm must pay a sunk cost of entry fe .
2 Uncertain lifetime if it chooses to enter.
Exogenous probability δ of ”death” i.e., a bad shock that will cause itto exit.HMY and Chaney (AER 2008) ignore the second source, assuming thata successful entrant produces for one period only.This simplifies the model a lot without affecting its main predictions,and many authors have followed them; but it is insightful to solve thefree entry case in full.
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 6 / 29
Overview of the Melitz Model Continuum CES Preferences
Continuum CES Preferences
CES preferences with a continuum of goods: little new:
U =[∫
ω∈Ω q (ω)θ dω]1/θ
,
0 < θ = σ−1σ < 1, σ = 1
1−θ > 1
⇒ Optimal consumption: q (ω) =[p(ω)P
]−σQ Q = U
Optimal expenditure: r (ω) ≡ p (ω) q (ω) =[p(ω)P
]1−σR
R = PQ =∫
ω∈Ω r (ω) dω
Price index: P =[∫
ω∈Ω p (ω)1−σ dω] 1
1−σ
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 7 / 29
Equilibrium in Autarky
Plan of Lectures
1 Empirical Background
2 Overview of the Melitz Model
3 Equilibrium in AutarkyProductionEntryEquilibrium Selection in Autarky: FigureProductivity of EntrantsAverage Productivity, Prices and ProfitsZCP (Zero Cutoff Profit) ConditionFE (Free Entry) ConditionIndustry Equilibrium: FigureEquilibrium in Autarky: RecapGeneral Equilibrium in Autarky: Details
4 Effects of Trade
5 Extensions
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 8 / 29
Equilibrium in Autarky Production
Production
Firms have different productivities ϕ (inverse of variable costs c).
All infra-marginal firms make positive profits; otherwise little new:
Labour the only factor, with w = 1: TC (c) = f + cq (c)
Profit maximisation implies: p (c) = σσ−1 c , q (c) =
(σ
σ−1 c)−σ
Pσ−1R
Price-cost margin: p (c)− c = 1σ−1 c = 1
σ p (c)
Revenue: r (c) =(
σσ−1 c
)1−σPσ−1R =
(σ−1
σ1c P)σ−1
R
Variable profit:
r (c)− cq (c) = [p (c)− c ] q (c) = 1σ r (c) =
(σ−1
σ1c P)σ−1
Rσ
Total profit: π (c) = 1σ r (c)− f =
(σ−1
σ1c P)σ−1
Rσ − f
Higher productivity firms have higher output, revenue and profits.They also charge lower prices.Ratios (rankings) of p, q, and r depend only on productivities:
p(c1)p(c2)
= c1c2
q(c1)q(c2)
=(c2c1
)σand
r (c1)r (c2)
=(c2c1
)σ−1
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 9 / 29
Equilibrium in Autarky Entry
Entry
To be or not to be? Depends on firm’s expected value given ϕ:
V (ϕ) = max
0,
∞∑0(1− δ)t π (ϕ)
= max
0, 1
δ π (ϕ)
.
This is steady-state analysis; no learning-by-doing etc.Probability of death acts just like a discount factor.
So: Entry occurs IFF V (ϕ) ≥ 0⇔ π (ϕ) ≥ 0⇔ ϕ ≥ ϕ∗ where:
ϕ∗ : π (ϕ∗) = 0 (1)
So far, very like homogeneous-firms model; but there:π = 0 holds for all firms (since they are homogeneous);This pins down q for all firms
... and market-clearing pins down mass of firms n.
Here: π (ϕ∗) = 0 is one equation in ϕ∗ and (through P) n.It determines q (ϕ∗) but there are many other q (ϕ) ...So: we need more equations ...
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 10 / 29
Equilibrium in Autarky Equilibrium Selection in Autarky: Figure
Equilibrium Selection in Autarky
c
11 c1*)(c
f
EnterExit EnterExit
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 11 / 29
Equilibrium in Autarky Productivity of Entrants
Productivity of Entrants
Equilibrium distribution of firm productivities:
Ex ante, it continues to be g (ϕ).Ex post, distribution of entrants’ productivities is different: µ (ϕ).Probability of a bad draw is G (ϕ∗); so probability of entry is1− G (ϕ∗)Hence distribution of ϕ on [ϕ∗, ∞), conditional on successful entry, is:
µ (ϕ) =
g (ϕ)
1−G (ϕ∗) if ϕ ≥ ϕ∗
0 otherwise(2)
Aggregate price: P =[∫ ∞
0 p (ϕ)1−σ Mµ (ϕ) d ϕ] 1
1−σ
= M1
1−σ p (ϕ) M: mass of firms.
where: ϕ (ϕ∗) ≡[∫ ∞
ϕ∗ ϕσ−1µ (ϕ) d ϕ] 1
σ−1
Average productivity; (strictly, a ”weighted symmetric mean”).
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 12 / 29
Equilibrium in Autarky Average Productivity, Prices and Profits
Average Productivity, Prices and Profits
Proof that P = M1
1−σ p (ϕ)
P =[∫
ω∈Ω p (ω)1−σ dω] 1
1−σ=[∫ ∞
0 p (ϕ)1−σ Mµ (ϕ) d ϕ] 1
1−σ
Recall:p(ϕ1)p(ϕ2)
= ϕ2ϕ1
so: p (ϕ) = p (ϕ)ϕϕ
⇒ P = M1
1−σ
[∫ ∞0 p (ϕ)1−σ ϕ1−σ
ϕ1−σ µ (ϕ) d ϕ] 1
1−σ
= M1
1−σ p (ϕ) ϕ[∫ ∞
0 ϕσ−1µ (ϕ) d ϕ] 1
1−σ
= M1
1−σ p (ϕ) from defn. of ϕ (Careful with exponents!) QED
ϕ is a ”sufficient statistic” for the industry.In particular: Average profits π = π (ϕ);
Proof: π ≡∫ ∞
ϕ∗ π (ϕ) µ (ϕ) d ϕ; BUT: π (ϕ) = 1σ r (ϕ)− f
= 1σ
∫ ∞ϕ∗ r (ϕ) µ (ϕ) d ϕ− f ; BUT:
r (ϕ)r (ϕ)
=(
ϕϕ
)σ−1
= 1σ r (ϕ)
(1ϕ
)σ−1 ∫ ∞ϕ∗ ϕσ−1µ (ϕ) d ϕ− f ; BUT: Recall defn. of ϕ.
= 1σ r (ϕ)− f QED
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 13 / 29
Equilibrium in Autarky ZCP (Zero Cutoff Profit) Condition
ZCP (Zero Cutoff Profit) Condition
Given π = 1σ r (ϕ)− f and ϕ (ϕ∗), π is a function of ϕ∗.
It’s even neater than that, recalling thatr (ϕ)r (ϕ∗) =
[ϕ(ϕ∗)
ϕ∗
]σ−1;
⇒ π = 1σ r (ϕ∗)
[ϕ(ϕ∗)
ϕ∗
]σ−1− f ; BUT: π (ϕ∗) = 1
σ r (ϕ∗)− f = 0;
⇒ π =
[ϕ (ϕ∗)
ϕ∗
σ−1
− 1
]f [ZCP] (3)
Slope in π, ϕ∗ space depends on two competing effects:1 Higher ϕ∗ raises average productivity, and so profits, of surviving firms:
ϕ′ (ϕ∗) > 0.A selection effect, not a firm-level productivity effect
2 Higher ϕ∗ means tougher competition: profits are decreasing in rivals’productivity.
(2) dominates for many distributions: ZCP downward-sloping.e.g., fat-enough tails: lognormal, exponential, etc.
(1) and (2) exactly cancel for Pareto: G (ϕ) = 1−(
bϕ
)k: ZCP flat.
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 14 / 29
Equilibrium in Autarky FE (Free Entry) Condition
FE (Free Entry) Condition
Expected PV of profits must equal sunk cost of entry:∫ ∞0 v (ϕ) g (ϕ) d ϕ = fe ; BUT: v (ϕ) = 1
δ π (ϕ) for ϕ ≥ ϕ∗;otherwise = 0;⇒ 1
δ
∫ ∞ϕ∗ π (ϕ) g (ϕ) d ϕ = fe ; BUT: Recall distribution of entrants’
productivities;⇒ 1
δ [1− G (ϕ∗)]∫ ∞
ϕ∗ π (ϕ) µ (ϕ) d ϕ = fe ; BUT: Recall defn. of π;
⇒ π =δfe
1− G (ϕ∗)[FE] (4)
So: Average industry profits rise with δ (think Grim Reaper), fe (thinklarger firms) and ϕ∗ (higher productivity cutoff).
(3) and (4): Two equations in two unknowns, π and ϕ∗;
So they are determined by fe , f , δ, and g (ϕ) only.
Melitz shows that FE must be cut by ZCP only once from above;
So equilibrium is unique and stable.
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 15 / 29
Equilibrium in Autarky Industry Equilibrium: Figure
Industry Equilibrium in Autarky
FE
ef
ZCP
*
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 16 / 29
Equilibrium in Autarky Equilibrium in Autarky: Recap
Equilibrium in Autarky: Recap
The story so far:
p (ϕ) = σσ−1
1ϕ P = M
11−σ p (ϕ) = M
11−σ σ
σ−11ϕ
π (ϕ) = 1σ r (ϕ)− f π = 1
σ r (ϕ)− f
r (ϕ) =(
σ−1σ ϕP
)σ−1R
r (ϕ) =(
σ−1σ ϕP
)σ−1R =
(M
11−σ
)σ−1R = R
M
This implies that: π = 1σRM − f
σ, f given; π and ϕ∗ determined by (3) and (4).
Finally: R determined by aggregate budget constraint: R = wL = L.
N.B. This is the first, and only, place where GE appears: see next page.BUT: it is crucial: without induced rise in real wage in GE, theselection effects of trade would not arise in the model.
Only remaining unknown is M, which is therefore: M = L(π+f )σ .
Compare Krugman: M = Lcy+f from LME; = L
f σ from CES.
Since y = (σ− 1) fc → cy + f = (σ− 1)f − f .
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 17 / 29
Equilibrium in Autarky General Equilibrium in Autarky: Details
General Equilibrium in Autarky: Details
Stationary equilibrium: Aggregate variables stay constant.
So, mass of new entrants each period Me must be such that mass ofsuccessful entrants equals mass of exiters: [1− G (ϕ∗)]Me = δM.Successful entrants and failing incumbents have the same productivitydistribution;So: equilibrium distribution µ (ϕ) is not affected by simultaneous entryand exit.
Finally, how come R = L? What happened to profits?
The trick is that sunk entry costs also require labour.So: L = Lp + Le ; aggregate labour used for production and investment(in entry).But: wLp = R −Π;
while: wLe = wMe fe = w δM1−G (ϕ∗) fe = Mπ = Π
So, with w = 1, L = R −Π + Π = R.
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 18 / 29
Effects of Trade
Plan of Lectures
1 Empirical Background
2 Overview of the Melitz Model
3 Equilibrium in Autarky
4 Effects of TradeTrade CostsFirms in Home and Export MarketsEquilibrium Selection in Trade: FigureThe ZCP Locus with TradeIndustry Equilibrium in Autarky and Trade: FigureAdjustment to Trade LiberalisationComparing Trade and AutarkyComparing Trade and Autarky (cont.)
5 ExtensionsJ.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 19 / 29
Effects of Trade Trade Costs
Trade Costs
Now: replicate the home economy without trade costs:
n: # foreign countries; all identical, so FPE holds, w = 1 in all.Budget constraint is now: R = (n + 1) L.All firms will export, trade does not affect average productivity.(3) and (4) continue to determine the same equilibrium π and ϕ∗.
[Think Krugman 1979: n + 1 > 0 ⇒ M = n + 1.]
So, we need trade costs; 2 kinds in fact:
1 Iceberg variable costs: τ2 Fixed cost of exporting fx ; incurred after firm learns its ϕ.
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 20 / 29
Effects of Trade Firms in Home and Export Markets
Firms in Home and Export Markets
Firms charge constant mark-ups in both domestic and exportmarkets.
So: Domestic revenue: rd (ϕ) =(
σ−1σ ϕP
)σ−1R as before
Revenue in export market j : rX (ϕ) = τ1−σ(
σ−1σ ϕPj
)σ−1Rj
Symmetry ⇒ Total exporter revenue:(1 + nτ1−σ
)rd (ϕ)
Profits on domestic sales: πd (ϕ) =rd (ϕ)
σ − f .This = 0 determines threshold ϕ∗ as before.
Profits in an export market: πX (ϕ) =rX (ϕ)
σ − fX = τ1−σrd (ϕ)σ − fX .
This = 0 determines a new threshold ϕ∗X .
Threshold ratio:rd(ϕ∗X )rd (ϕ∗) = τσ−1 fX
f BUT:rd(ϕ∗X )rd (ϕ∗) =
(ϕ∗Xϕ∗
)σ−1
⇒ ϕ∗X = τ
(fXf
) 1σ−1
ϕ∗ (5)
i.e., sorting as in data (ϕ∗X > ϕ∗) IFF τ(fXf
) 1σ−1
> 1⇔ τσ−1fX > f .
We assume this holds from now on. (A little unsatisfactory ... )
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 21 / 29
Effects of Trade Equilibrium Selection in Trade: Figure
Equilibrium Selection in Trade
c
cx
11 c1*)(c 1* )( xc )( x
f
f ExportHome SalesOnlyxf
EnterExit
pOnly
EnterExit
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 22 / 29
Effects of Trade The ZCP Locus with Trade
The ZCP Locus with Trade
Probability of entry = 1− G (ϕ∗) as before;
Probability that an entrant exports: pX =1−G(ϕ∗X )1−G (ϕ∗)
Average expected profits of an entrant:π = πd (ϕ) + pX nπX (ϕX )
Here: ϕX is the weighted average productivity of exporters; so:
π(ϕ∗)=
[ϕ(ϕ∗)
ϕ∗
σ−1
−1
]f +pXn
[ϕX (ϕ∗)ϕ∗X (ϕ∗)
σ−1
−1
]fX [ZCPt ] (6)
This is clearly greater than in autarky; i.e., π (ϕ∗) > πa (ϕ∗) for anyarbitrary ϕ∗.i.e., ZCP shifts up relative to autarky.Finally: FE locus is unaffected.So: ϕ∗ > ϕ∗a and π (ϕ∗) > πa (ϕ∗a)
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 23 / 29
Effects of Trade Industry Equilibrium in Autarky and Trade: Figure
Industry Equilibrium in Autarky and Trade
FE)( *a
*
)( a
)( *t
ZCP - Trade
)( *aaa
efZCP - Autarkyy
**a
*t
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 24 / 29
Effects of Trade Adjustment to Trade Liberalisation
Adjustment to Trade Liberalisation
Result that ϕ∗ > ϕ∗a matches the data:
Trade causes marginal firms to exit;Selection effect of trade;Why? NOT a competition effect through demand side.
Remember: CES: Firm size and therefore π fixed by costs.Ans.: Labour market adjustment is crucial (though in the background).Increase in profitable opportunities for relatively more productive firms⇒ more entry⇒ Increase in labour demand⇒ Increase in real wage w
P ; i.e. fall in P⇒ Least productive firms in autarky can no longer make profits.
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 25 / 29
Effects of Trade Comparing Trade and Autarky
Comparing Trade and Autarky
Mass of active home firms falls: M < Ma; proof:
Total revenue of domestic producers: R = LAverage revenue: r =
∫ ∞0 r (ϕ) µ (ϕ) d ϕ
= σ (π + f + pX nfX )
BUT: R = Mr ⇒ M = Lσ(π+f +pXnfX )
< Ma
(Recall that Ma = Lσ(πa+f )
and π > πa.
What about total number of varieties?
# firms selling in any one market: Mt = (1 + npX )M
Likely (except for very high τ) that Mt > Ma
i.e., gains from variety.
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 26 / 29
Effects of Trade Comparing Trade and Autarky (cont.)
Comparing Trade and Autarky (cont.)
ϕ∗ > ϕ∗a ⇒ rd (ϕ) = σ(
ϕϕ∗
)σ−1f < ra (ϕ) = σ
(ϕϕ∗a
)σ−1f
i.e., all firms earn less on home market.However: ra (ϕ) < rd (ϕ) + nrX (ϕ) for ϕ > ϕ∗X (harder to prove ... )i.e., exporting firms gain.
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 27 / 29
Extensions
Plan of Lectures
1 Empirical Background
2 Overview of the Melitz Model
3 Equilibrium in Autarky
4 Effects of Trade
5 Extensions
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 28 / 29
Extensions
Extensions
Quadratic preferences ⇒ Variable mark-ups, competition effects:Melitz-Ottaviano (REStud 2007); but, partial equilibrium.
Combine with HO: Bernard-Redding-Schott (REStud 2008).
Quality: Baldwin-Harrigan (AEJ 2011) and many more:Predict that price rises not falls with productivity.
Zeroes in the Trade Matrix: Helpman-Melitz-Rubinstein (QJE 2008):Link with gravity model; avoids prediction that every firm serves everyexport market.
Other types of sorting:FDI: HMY (AER 2004)R&D: Bustos (AER 2011)Wages: Egger-Kreickemeier (IER 2009), Helpman-Itskhoki-Redding(Em 2010).In all cases: Trade-off between fixed and variable costs;
“Only more productive firms select into higher fixed-cost activity.”Or, is that true? See next file and Mrazova-Neary (2012) . . .
J.P. Neary (University of Oxford) Heterogeneous Firms January 30, 2013 29 / 29